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The convergence of a Galerkin approximation scheme for an extensible beam. (English) Zbl 0727.73093
The equation governing the transverse displacement of an extensible beam with hinged ends is treated by a semi-discrete Galerkin approximate scheme. The rate of convergence and error estimates are discussed. A fully discrete scheme applying Crank-Nicolson time discretization is also discussed.

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
74H45 Vibrations in dynamical problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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References:
[1] [1] G. A. BAKER & J. H. BRAMBLE, Semidiscrete and single step fully discrete approximations for second order hyperbolic equations, RAIRO Anal. Numer. 13 (1979), 75-100. Zbl0405.65057 MR533876 · Zbl 0405.65057
[2] J. M. BALL, Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl. 42 (1973), 61-90. Zbl0254.73042 MR319440 · Zbl 0254.73042
[3] I. CHRISTIE & J. M. SANZ-SERNA, A Galerkin method for a nonlinear integro-differential wave system, Comp. Meth. Appl. Mech. Eng. 44 (1984), 229-237. Zbl0525.73089 MR757058 · Zbl 0525.73089
[4] R. COURANT & D. HILBERT, Methods of Mathematical Physics, Vol. 1, Wiley-Interscience, New York, 1953. Zbl0051.28802 MR65391 · Zbl 0051.28802
[5] R. W. DICKEY, Free vibrations and dynamic buckling of an extensible beam, Math. Anal. Appl. 29 (1970), 443-454. Zbl0187.04803 MR253617 · Zbl 0187.04803
[6] T. GEVECI, On the convergence of Galerkin approximation schemes for second-order hyperbolic equations in energy and negative norms, Math. Compt. 42 (1984), 393-415. Zbl0553.65082 MR736443 · Zbl 0553.65082
[7] P. HOLMES & J. MARSDEN, Bifurcation to divergence and flutter in flow-induced oscillations : An infinite dimensional analysis, Automatica 14 (1978), 367-384. Zbl0385.93028 MR495662 · Zbl 0385.93028
[8] J. RAUCH, On convergence of the finite element method for the wave equation, SIAM J. Numer. Anal. 22 (1985), 245-249. Zbl0575.65091 MR781318 · Zbl 0575.65091
[9] J. M. SANZ-SERNA, Methods for the numerical solution of the nonlinear Schroedinger equation, Math. Compt. 43 (1984), 21-27. Zbl0555.65061 MR744922 · Zbl 0555.65061
[10] G. STRANG & G. J. FIX, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, N.J., 1973. Zbl0356.65096 MR443377 · Zbl 0356.65096
[11] V. THOMÉE, Negative norm estimates and superconvergence in Galerkin methods for parabolic problems, Math. Compt. 34 (1980), 99-113. Zbl0454.65077 MR551292 · Zbl 0454.65077
[12] V. THOMÉE, Galerkin Finite Element Methods for Parabolic Problems, Springer lecture Notes in Mathematics v. 1054, Springer-Verlag, Berlin, 1984. Zbl0528.65052 MR744045 · Zbl 0528.65052
[13] S. WOINOWSKY-KRIEGER, The effect of the axial force on the vibration of hinged bars, J. Appl. Mech, 17 (1950), 35-36. Zbl0036.13302 MR34202 · Zbl 0036.13302
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